An estimate of Green's function of the problem of bounded solutions in the case of a triangular coefficient

TL;DR

This work studies the Green's function for the bounded solutions problem $x'(t)=A x(t)+f(t)$ with triangularizable coefficient matrices. It derives an explicit bound on $\|\\mathcal{G}(B,t)\\|$ for $B=D+N$ by exploiting a convolution-based expansion and a Fourier-analysis expression of the k-fold convolution, yielding a bound in terms of $\\|N\\|$, $t$, and the parameters $\\gamma^-,\\gamma^+$ that separate the spectrum from the imaginary axis. In the special case $N=0$ the bound reduces to Van Loan's classic estimate for the matrix exponential, and the method extends to entrywise bounds under $|\\cdot|$-norms. The results provide practical, computable estimates of Green's functions that are useful for numerical analysis of linear ODE systems in Schur form or with triangular coefficients.

Abstract

An estimate of Green's function of the bounded solutions problem for the ordinary differential equation $x'(t)-Bx(t)=f(t)$ is proposed. It is assumed that the matrix coefficient $B$ is triangular. This estimate is a generalization of the estimate of the matrix exponential proved by Ch. F. Van Loan.

Theorems & Definitions (14)

• Theorem 1: Daletskii-Krein:eng
• Corollary 2
• proof
• Proposition 3
• proof
• Theorem 4
• Remark 1
• Remark 2
• Remark 3
• proof : Proof of Theorem \ref{['t:fin']}